Chapter 1 Integers (Concepts)

Integers & Their Addition and Subtraction

In earlier classes, you were introduced to the world of numbers, starting with natural numbers ($1, 2, 3, ...$) used for counting, and then expanding to whole numbers ($0, 1, 2, 3, ...$) by including zero. Now, we will broaden our number system even further to include a new set of numbers alongside whole numbers. This larger collection is called Integers. Integers are essential for representing quantities that can be opposite in direction or value, such as temperatures above and below zero, or gaining and losing money.

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What are Integers?

Integers are a collection of numbers that includes all whole numbers ($0, 1, 2, 3, ...$) and the negative counterparts of the natural numbers ($-1, -2, -3, ...$). This means integers can be positive, negative, or zero.

The set of integers can be represented as: $\{..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ...\}$.

Integers are denoted by the symbol $\mathbb{Z}$ (or sometimes $I$). The symbol $\mathbb{Z}$ comes from the German word 'Zahlen', which means 'numbers'.

Based on their value relative to zero, integers are classified into three groups:

• Positive Integers: These are the integers greater than zero. They are $1, 2, 3, 4, 5, ...$. These are the same as the natural numbers. A positive sign ($+$) is often not written before a positive integer (e.g., $+5$ is usually written as $5$).
• Negative Integers: These are the integers less than zero. They are $..., -5, -4, -3, -2, -1$. A negative sign ($-$) is always written before a negative integer.
• Zero: The integer $0$ is neither positive nor negative. It serves as the dividing point between positive and negative integers on the number line.

All natural numbers are positive integers. All whole numbers are integers. However, not all integers are whole numbers (e.g., $-7$ is an integer but not a whole number), and not all integers are natural numbers (e.g., $0$ and $-7$ are integers but not natural numbers).

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Representation of Integers on a Number Line

The number line is a valuable tool for visualising and comparing integers. It helps us understand the order and relative values of integers.

To represent integers on a number line, follow these steps:

1. Draw a straight line and extend it infinitely in both directions (indicated by arrows at the ends).
2. Mark a point near the center of the line and label it as $0$. This is the origin.
3. Mark points to the right of $0$ at equal distances. Label these points $1, 2, 3, 4, ...$ in increasing order. These represent the positive integers. The direction to the right of $0$ is the positive direction.
4. Mark points to the left of $0$ at the same equal distances as used on the right. Label these points $-1, -2, -3, -4, ...$ in decreasing order. These represent the negative integers. The direction to the left of $0$ is the negative direction.

From the number line, we can make the following observations about comparing integers:

• As you move to the right on the number line, the value of the integers increases.
• As you move to the left on the number line, the value of the integers decreases.
• Any integer on the right of another integer is greater than the integer on the left. For example, $3 > 1$ (3 is to the right of 1), $0 > -4$ (0 is to the right of -4), $-2 > -5$ (-2 is to the right of -5).
• Every positive integer is greater than zero.
• Every negative integer is less than zero.
• Every positive integer is greater than every negative integer.

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Absolute Value of an Integer

The Absolute Value of an integer is its numerical value without considering whether it is positive or negative. It represents the distance of the integer from zero on the number line. Since distance is always measured as a non-negative quantity (you can't walk a negative distance), the absolute value of any integer is always non-negative (zero or a positive number).

The absolute value of an integer 'a' is denoted by writing the integer within two vertical bars, like this: $|a|$.

For example:

• $|5| = 5$. The distance of 5 from 0 on the number line is 5 units.
• $|-5| = 5$. The distance of -5 from 0 on the number line is also 5 units.
• $|0| = 0$. The distance of 0 from 0 is 0 units.

We can define the absolute value using a piecewise function:

When $a$ is negative, $-a$ represents the additive inverse of $a$, which is its positive counterpart. For instance, if $a = -8$, then $|a| = |-8| = -(-8) = 8$.

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Addition of Integers

Adding integers involves combining quantities that might be positive (moving right on the number line) or negative (moving left on the number line). We can formulate rules for addition based on the signs of the integers being added.

Rule 1: Adding two positive integers

When adding two positive integers, the sum is always a positive integer. Simply add their absolute values (which are the numbers themselves) and the result is positive.

$(+\text{a}) + (+\text{b}) = +(\text{a} + \text{b})$ or $a + b$.

Example: $(+5) + (+3) = 5 + 3 = 8$. (On number line: Start at 5, move 3 units right $\to$ 8)

Example: $10 + 15 = 25$.

Rule 2: Adding two negative integers

When adding two negative integers, the sum is always a negative integer. Add their absolute values and place a negative sign before the sum.

$(-\text{a}) + (-\text{b}) = -(\text{a} + \text{b})$.

Example: $(-5) + (-3)$. Absolute values are $|-5|=5$ and $|-3|=3$. Sum of absolute values is $5+3=8$. Place a negative sign: $-8$.

(On number line: Start at -5, move 3 units left $\to$ -8)

Example: $(-10) + (-15) = -(10+15) = -25$.

Rule 3: Adding one positive and one negative integer

When adding a positive integer and a negative integer, the sign of the sum depends on which integer has the larger absolute value. To find the sum, find the difference between their absolute values. The sum takes the sign of the integer with the greater absolute value.

$(+\text{a}) + (-\text{b})$ or $(-\text{a}) + (+\text{b})$

Result is $(|a| - |b|)$ or $(|b| - |a|)$ with the sign of the integer having the greater absolute value.

Example 1: $(+7) + (-3)$. Absolute values are $|+7|=7$ and $|-3|=3$. Difference $7-3=4$. Integer with larger absolute value is $+7$ (positive). Sum is $+4$ or $4$.

(On number line: Start at 7, move 3 units left $\to$ 4)

Example 2: $(-7) + (+3)$. Absolute values are $|-7|=7$ and $|+3|=3$. Difference $7-3=4$. Integer with larger absolute value is $-7$ (negative). Sum is $-4$.

(On number line: Start at -7, move 3 units right $\to$ -4)

Example 3: $(+4) + (-4)$. Absolute values are $|+4|=4$ and $|-4|=4$. Difference $4-4=0$. The sum is $0$. (+4 and -4 are additive inverses, their sum is always 0).

Example 4: $(-10) + (+12)$. Absolute values $|-10|=10, |+12|=12$. Difference $12-10=2$. Sign of $+12$ (positive). Sum is $+2$ or $2$.

Example 5: $(+10) + (-12)$. Absolute values $|+10|=10, |-12|=12$. Difference $12-10=2$. Sign of $-12$ (negative). Sum is $-2$.

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Subtraction of Integers

Subtraction of integers is defined in terms of addition. Subtracting an integer is the same as adding its additive inverse (opposite).

The Additive Inverse of an integer 'a' is the integer that, when added to 'a', gives zero. The additive inverse of 'a' is denoted by '-a'.

• Additive inverse of $5$ is $-5$, because $5 + (-5) = 0$.
• Additive inverse of $-3$ is $+3$ (or 3), because $(-3) + (+3) = 0$. So, $-(-3) = 3$.
• Additive inverse of $0$ is $0$, because $0+0=0$.

Rule for Subtraction: To subtract an integer 'b' from an integer 'a', add the additive inverse of 'b' to 'a'.

Since the additive inverse of $b$ is $-b$, the rule is $a - b = a + (-b)$.

Since the additive inverse of $-b$ is $b$, the rule for subtracting a negative is $a - (-b) = a + (\text{additive inverse of } -b) = a + b$.

Examples of Subtraction:

1. Subtract $(+3)$ from $(+5)$. This is written as $(+5) - (+3)$.

$= (+5) + (-3)$

Now, use the addition rule for $(+\text{a}) + (-\text{b})$: $|+5|=5, |-3|=3$. Difference $5-3=2$. Sign is positive (from +5). Result is 2.

2. Subtract $(-3)$ from $(+5)$. This is written as $(+5) - (-3)$.

$= (+5) + (+3)$

Now, use the addition rule for two positive integers: $5+3=8$. Result is 8.

3. Subtract $(+3)$ from $(-5)$. This is written as $(-5) - (+3)$.

$= (-5) + (-3)$

Now, use the addition rule for two negative integers: $-(5+3)=-8$. Result is -8.

4. Subtract $(-3)$ from $(-5)$. This is written as $(-5) - (-3)$.

$= (-5) + (+3)$

Now, use the addition rule for $(-\text{a}) + (+\text{b})$: $|-5|=5, |+3|=3$. Difference $5-3=2$. Sign is negative (from -5). Result is -2.

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Addition and Subtraction of Integers on a Number Line

The number line provides a visual way to perform addition and subtraction of integers. The basic rules of movement are:

• Adding a positive integer: Move to the right.
• Adding a negative integer: Move to the left.

Since subtraction is defined as adding the opposite, the rules for subtraction are:

• Subtracting a positive integer ($a - b = a + (-b)$): Move to the left.
• Subtracting a negative integer ($a - (-b) = a + b$): Move to the right.

Examples of Addition on the Number Line

(a) Find $2 + 3$:

Start at $2$. Since we are adding a positive integer ($3$), move $3$ units to the right. We land on $5$. So, $2 + 3 = 5$.

(b) Find $(-2) + (-3)$:

Start at $-2$. Since we are adding a negative integer ($-3$), move $3$ units to the left. We land on $-5$. So, $(-2) + (-3) = -5$.

(c) Find $5 + (-3)$:

Start at $5$. Since we are adding a negative integer ($-3$), move $3$ units to the left. We land on $2$. So, $5 + (-3) = 2$.

Examples of Subtraction on the Number Line

(a) Find $4 - 6$:

This is the same as $4 + (-6)$. Start at $4$. Move $6$ units to the left. We land on $-2$. So, $4 - 6 = -2$.

(b) Find $3 - (-4)$:

This is the same as $3 + 4$. Start at $3$. Move $4$ units to the right. We land on $7$. So, $3 - (-4) = 7$.

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Properties of Addition and Subtraction of Integers

When we work with numbers, understanding their properties helps us perform calculations more efficiently and provides a deeper insight into how they behave. Integers, as an extended set of numbers, also follow certain properties under the operations of addition and subtraction. Let's explore these important properties.

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1. Closure Property

The closure property states that when you perform an operation on two numbers from a set, the result is also a number belonging to the same set.

(a) Closure under Addition

The set of integers is closed under addition. This means that if you add any two integers, the sum you get will always be another integer.

In mathematical terms: If '$a$' and '$b$' are any two integers, then their sum, $a + b$, is also an integer.

Let's look at some examples:

• Adding two positive integers: $5 + 3 = 8$. Here, 5, 3, and 8 are all integers.
• Adding two negative integers: $(-5) + (-3) = -8$. Here, -5, -3, and -8 are all integers.
• Adding a positive and a negative integer: $5 + (-3) = 2$. Here, 5, -3, and 2 are all integers.
• Adding a negative and a positive integer: $(-5) + 3 = -2$. Here, -5, 3, and -2 are all integers.

In every case, adding two integers results in an integer. So, the set of integers is closed under addition.

(b) Closure under Subtraction

The set of integers is also closed under subtraction. This means that if you subtract one integer from another integer, the difference you get will always be an integer.

In mathematical terms: If '$a$' and '$b$' are any two integers, then their difference, $a - b$, is also an integer.

Let's look at some examples:

• Subtracting positive from positive: $7 - 3 = 4$. Here, 7, 3, and 4 are all integers.
• Subtracting negative from negative: $(-7) - (-3) = (-7) + 3 = -4$. Here, -7, -3, and -4 are all integers.
• Subtracting negative from positive: $7 - (-3) = 7 + 3 = 10$. Here, 7, -3, and 10 are all integers.
• Subtracting positive from negative: $(-7) - 3 = (-7) + (-3) = -10$. Here, -7, 3, and -10 are all integers.

In every case, subtracting one integer from another results in an integer. So, the set of integers is closed under subtraction.

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2. Commutative Property

The commutative property deals with whether the order of the numbers matters in an operation.

(a) Commutativity of Addition

Addition of integers is commutative. This means that when you add two integers, changing the order in which you add them does not change the sum.

In mathematical terms: If '$a$' and '$b$' are any two integers, then $a + b = b + a$.

Let's verify this with an example:

Let $a = 5$ and $b = -3$.

Calculate $a + b$:

$a + b = 5 + (-3)$

Using the rule for adding a positive and negative integer ($|5|=5, |-3|=3$, difference $5-3=2$, sign of 5 is positive):

$5 + (-3) = 2$

Calculate $b + a$:

$b + a = (-3) + 5$

Using the rule for adding a negative and positive integer ($|-3|=3, |5|=5$, difference $5-3=2$, sign of 5 is positive):

$(-3) + 5 = 2$

Since both $a+b$ and $b+a$ give the same result (2), the commutative property holds for addition of integers: $5 + (-3) = (-3) + 5$.

(b) Subtraction is NOT Commutative for Integers

Subtraction of integers is not commutative. This means that if you change the order of the integers in a subtraction, the result generally changes (unless the two integers are equal).

In mathematical terms: If '$a$' and '$b$' are any two distinct integers ($a \neq b$), then $a - b \neq b - a$.

Let's verify this with an example:

Let $a = 5$ and $b = -3$.

Calculate $a - b$:

$a - b = 5 - (-3)$

Using the subtraction rule ($a - (-b) = a + b$):

$5 - (-3) = 5 + 3 = 8$

Calculate $b - a$:

$b - a = (-3) - 5$

Using the subtraction rule ($a - b = a + (-b)$):

$(-3) - 5 = (-3) + (-5)$

Using the rule for adding two negative integers (add absolute values and make negative):

$(-3) + (-5) = -(3+5) = -8$

Since $8 \neq -8$, the commutative property does not hold for subtraction of integers: $5 - (-3) \neq (-3) - 5$.

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3. Associative Property

The associative property deals with how grouping affects the result when performing an operation on three or more numbers.

(a) Associativity of Addition

Addition of integers is associative. This means that when adding three or more integers, the way you group them using parentheses does not affect the final sum.

In mathematical terms: If '$a$', '$b$', and '$c$' are any three integers, then $(a + b) + c = a + (b + c)$.

Let's verify this with an example:

Let $a = -5$, $b = 3$, and $c = -2$.

Calculate the Left Hand Side (LHS): $(a + b) + c$

$(a + b) + c = ((-5) + 3) + (-2)$

First, calculate the sum inside the first parenthesis: $(-5) + 3$. (Rule 3: $|-5|=5, |3|=3$, difference 2, sign of -5). $(-5)+3 = -2$.

Now, add the result to $c$: $(-2) + (-2)$. (Rule 2: add absolute values and make negative). $-(2+2) = -4$.

So, LHS $= -4$.

Calculate the Right Hand Side (RHS): $a + (b + c)$

$a + (b + c) = (-5) + (3 + (-2))$

First, calculate the sum inside the parenthesis: $3 + (-2)$. (Rule 3: $|3|=3, |-2|=2$, difference 1, sign of 3). $3+(-2) = 1$.

Now, add $a$ to the result: $(-5) + 1$. (Rule 3: $|-5|=5, |1|=1$, difference 4, sign of -5). $(-5)+1 = -4$.

So, RHS $= -4$.

Since LHS = RHS (both are -4), the associative property holds for addition of integers: $(a + b) + c = a + (b + c)$.

(b) Subtraction is NOT Associative for Integers

Subtraction of integers is not associative. This means that the way you group integers when performing multiple subtractions affects the result.

In mathematical terms: If '$a$', '$b$', and '$c$' are any three integers, then $(a - b) - c \neq a - (b - c)$ (except for some specific cases where the equality might accidentally hold).

Let's verify this with an example:

Let $a = 8$, $b = 4$, and $c = 2$.

Calculate the Left Hand Side (LHS): $(a - b) - c$

$(a - b) - c = (8 - 4) - 2$

First, calculate inside the parenthesis: $8 - 4 = 4$.

Now, subtract $c$: $4 - 2 = 2$.

So, LHS $= 2$.

Calculate the Right Hand Side (RHS): $a - (b - c)$

$a - (b - c) = 8 - (4 - 2)$

First, calculate inside the parenthesis: $4 - 2 = 2$.

Now, subtract this result from $a$: $8 - 2 = 6$.

So, RHS $= 6$.

Since LHS ($2$) is not equal to RHS ($6$), $(a - b) - c \neq a - (b - c)$. The associative property does not hold for subtraction of integers.

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4. Additive Identity

The Additive Identity is a special number that, when added to any number, does not change the value of that number. For integers, the additive identity is zero.

Zero ($0$) is the additive identity for integers. This means that adding $0$ to any integer leaves the integer unchanged.

In mathematical terms: If '$a$' is any integer, then $a + 0 = 0 + a = a$.

Examples:

• $7 + 0 = 7$
• $(-4) + 0 = -4$
• $0 + 15 = 15$
• $0 + (-100) = -100$

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5. Additive Inverse

For every integer, there exists another integer called its additive inverse. The Additive Inverse of an integer is the number which, when added to the integer, gives the additive identity (0) as the sum.

In mathematical terms: For every integer '$a$', there exists an integer '$-a$' such that $a + (-a) = (-a) + a = 0$. The integer '$-a$' is called the additive inverse of '$a$'.

• The additive inverse of $5$ is $-5$, because $5 + (-5) = 0$.
• The additive inverse of $-9$ is $9$ (or $+9$), because $(-9) + 9 = 0$. Note that the additive inverse of a negative number is a positive number. $-(-9) = 9$.
• The additive inverse of $0$ is $0$, because $0 + 0 = 0$.

The concept of additive inverse is fundamental to understanding subtraction of integers, as we saw earlier: $a - b = a + (-b)$. Subtracting an integer is equivalent to adding its additive inverse.

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Multiplication of Integers

We have learned how to add and subtract integers, including understanding their properties. Now, we will explore the operation of Multiplication of Integers. Just as multiplication of whole numbers is repeated addition (e.g., $3 \times 4$ means adding 3 groups of 4, or $4+4+4$), multiplication of integers also relates to repeated addition, but we must consider the signs of the numbers.

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Rules for Multiplication of Integers

The sign of the product of two integers depends on the signs of the integers being multiplied. Here are the rules:

1. Multiplication of two positive integers

The product of two positive integers is always a positive integer. This is the same as multiplying natural numbers or whole numbers.

Or simply: Positive $\times$ Positive $=$ Positive

Example: $(+4) \times (+5) = 4 \times 5 = 20$.

Example: $10 \times 6 = 60$.

2. Multiplication of two negative integers

The product of two negative integers is always a positive integer. To find the product, multiply their absolute values.

Or simply: Negative $\times$ Negative $=$ Positive

Example: $(-4) \times (-5)$.

Multiply the absolute values: $|-4| \times |-5| = 4 \times 5 = 20$.

Since both integers are negative, the product is positive.

Example: $(-10) \times (-3) = +(10 \times 3) = 30$.

3. Multiplication of one positive and one negative integer

The product of a positive integer and a negative integer (in any order) is always a negative integer. To find the product, multiply their absolute values and place a negative sign before the result.

Or simply: Positive $\times$ Negative $=$ Negative, and Negative $\times$ Positive $=$ Negative.

Example 1: $(+4) \times (-5)$.

Multiply the absolute values: $|+4| \times |-5| = 4 \times 5 = 20$.

Since one integer is positive and the other is negative, the product is negative.

Example 2: $(-4) \times (+5)$.

Multiply the absolute values: $|-4| \times |+5| = 4 \times 5 = 20$.

Since one integer is negative and the other is positive, the product is negative.

Example 3: $7 \times (-9) = -(7 \times 9) = -63$.

Example 4: $(-11) \times 2 = -(11 \times 2) = -22$.

Summary of Sign Rules for Multiplication:

The sign rules for multiplication can be summarised concisely:

First Integer Sign	Second Integer Sign	Product Sign
Positive (+)	Positive (+)	Positive (+)
Negative (-)	Negative (-)	Positive (+)
Positive (+)	Negative (-)	Negative (-)
Negative (-)	Positive (+)	Negative (-)

A simpler way to remember the sign rules: If the two integers have the **same sign**, the product is **positive**. If the two integers have **different signs**, the product is **negative**.

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Properties of Multiplication of Integers

Multiplication of integers follows several important properties, similar to addition, that are helpful for understanding and manipulating integer expressions.

1. Closure under Multiplication

The set of integers is closed under multiplication. This means that the product of any two integers is always an integer.

If '$a$' and '$b$' are any two integers, then their product, $a \times b$, is also an integer.

Examples:

• $6 \times 3 = 18$. Here, 6, 3, and 18 are all integers.
• $(-6) \times (-3) = 18$. Here, -6, -3, and 18 are all integers.
• $6 \times (-3) = -18$. Here, 6, -3, and -18 are all integers.

The closure property ensures that when you multiply any two integers, you will always get a result that is still within the set of integers.

2. Commutativity of Multiplication

Multiplication of integers is commutative. This means that the order in which you multiply two integers does not change the product.

In mathematical terms: If '$a$' and '$b$' are any two integers, then $a \times b = b \times a$.

Let's verify this with an example involving negative integers:

Let $a = -4$ and $b = 7$.

Calculate $a \times b$:

$a \times b = (-4) \times 7$

Using the rule for multiplying a negative and a positive integer:

$(-4) \times 7 = -(4 \times 7) = -28$.

Calculate $b \times a$:

$b \times a = 7 \times (-4)$

Using the rule for multiplying a positive and a negative integer:

$7 \times (-4) = -(7 \times 4) = -28$.

Since both $a \times b$ and $b \times a$ give the same result (-28), the commutative property holds for multiplication of integers: $(-4) \times 7 = 7 \times (-4)$.

3. Associativity of Multiplication

Multiplication of integers is associative. This property applies when multiplying three or more integers. It states that the way you group the integers using parentheses does not affect the final product.

In mathematical terms: If '$a$', '$b'$, and '$c$' are any three integers, then $(a \times b) \times c = a \times (b \times c)$.

Let's verify this with an example:

Let $a = -2$, $b = 3$, and $c = -4$.

Calculate the Left Hand Side (LHS): $(a \times b) \times c$

$(a \times b) \times c = ((-2) \times 3) \times (-4)$

First, calculate inside the first parenthesis: $(-2) \times 3$. (Negative $\times$ Positive $=$ Negative). $(-2) \times 3 = -6$.

Now, multiply the result by $c$: $(-6) \times (-4)$. (Negative $\times$ Negative $=$ Positive). $(-6) \times (-4) = +(6 \times 4) = 24$.

So, LHS $= 24$.

Calculate the Right Hand Side (RHS): $a \times (b \times c)$

$a \times (b \times c) = (-2) \times (3 \times (-4))$

First, calculate inside the parenthesis: $3 \times (-4)$. (Positive $\times$ Negative $=$ Negative). $3 \times (-4) = -(3 \times 4) = -12$.

Now, multiply $a$ by the result: $(-2) \times (-12)$. (Negative $\times$ Negative $=$ Positive). $(-2) \times (-12) = +(2 \times 12) = 24$.

So, RHS $= 24$.

Since LHS = RHS (both are 24), the associative property holds for multiplication of integers: $(a \times b) \times c = a \times (b \times c)$. This property allows us to multiply three or more integers by grouping any two first.

4. Multiplicative Identity

The Multiplicative Identity is a special number that, when multiplied by any number, results in the number itself. For integers, the multiplicative identity is one.

One ($1$) is the multiplicative identity for integers. Multiplying any integer by $1$ does not change its value.

In mathematical terms: If '$a$' is any integer, then $a \times 1 = 1 \times a = a$.

Examples:

• $9 \times 1 = 9$.
• $1 \times (-6) = -6$.
• $0 \times 1 = 0$.

Note: Multiplying an integer by $-1$ gives its additive inverse. If '$a$' is any integer, $a \times (-1) = (-1) \times a = -a$.

Example: $5 \times (-1) = -5$. The additive inverse of 5 is -5.

Example: $(-8) \times (-1) = 8$. The additive inverse of -8 is 8.

5. Property of Zero in Multiplication

The property of zero in multiplication of integers is the same as for whole numbers.

Rule: The product of any integer and zero ($0$) is always zero ($0$).

In mathematical terms: If '$a$' is any integer, then $a \times 0 = 0 \times a = 0$.

Examples:

• $15 \times 0 = 0$.
• $0 \times (-100) = 0$.
• $0 \times 0 = 0$.

6. Distributive Property of Multiplication over Addition

This property links the operations of multiplication and addition. It states that multiplying an integer by a sum of two other integers is the same as multiplying the integer by each of the other integers separately and then adding the products.

In mathematical terms: If '$a$', '$b$', and '$c$' are any three integers, then $a \times (b + c) = (a \times b) + (a \times c)$.

This property is extremely useful for simplifying calculations or rewriting expressions.

Let's verify this with an example:

Let $a = -3$, $b = 4$, and $c = -2$.

Calculate the Left Hand Side (LHS): $a \times (b + c)$

$a \times (b + c) = (-3) \times (4 + (-2))$

First, calculate the sum inside the parenthesis: $4 + (-2)$. (Rule 3 for addition: $|4|=4, |-2|=2$, difference 2, sign of 4). $4 + (-2) = 2$.

Now, multiply $a$ by the sum: $(-3) \times (2)$. (Rule 3 for multiplication: Negative $\times$ Positive $=$ Negative). $(-3) \times 2 = -6$.

So, LHS $= -6$.

Calculate the Right Hand Side (RHS): $(a \times b) + (a \times c)$

$(a \times b) + (a \times c) = ((-3) \times 4) + ((-3) \times (-2))$

First product: $(-3) \times 4$. (Negative $\times$ Positive $=$ Negative). $(-3) \times 4 = -12$.

Second product: $(-3) \times (-2)$. (Negative $\times$ Negative $=$ Positive). $(-3) \times (-2) = +6$.

Now, add the two products: $(-12) + (6)$. (Rule 3 for addition: $|-12|=12, |6|=6$, difference 6, sign of -12). $(-12) + 6 = -6$.

So, RHS $= -6$.

Since LHS = RHS (both are -6), the distributive property of multiplication over addition holds for integers: $a \times (b + c) = (a \times b) + (a \times c)$.

7. Distributive Property of Multiplication over Subtraction

Multiplication of integers also distributes over their subtraction.

In mathematical terms: If '$a$', '$b'$, and '$c$' are any three integers, then $a \times (b - c) = (a \times b) - (a \times c)$.

Let's verify this with an example:

Let $a = 5$, $b = 7$, and $c = 3$.

Calculate the Left Hand Side (LHS): $a \times (b - c)$

$a \times (b - c) = 5 \times (7 - 3)$

First, calculate the difference inside the parenthesis: $7 - 3 = 4$.

Now, multiply $a$ by the difference: $5 \times 4 = 20$.

So, LHS $= 20$.

Calculate the Right Hand Side (RHS): $(a \times b) - (a \times c)$

$(a \times b) - (a \times c) = (5 \times 7) - (5 \times 3)$

First product: $5 \times 7 = 35$.

Second product: $5 \times 3 = 15$.

Now, subtract the second product from the first: $35 - 15 = 20$.

So, RHS $= 20$.

Since LHS = RHS (both are 20), the distributive property of multiplication over subtraction holds for integers: $a \times (b - c) = (a \times b) - (a \times c)$.

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Multiplication of Three or More Negative Integers

When you multiply more than two integers, you can extend the sign rules. If there are multiple negative signs, the sign of the final product depends on whether the number of negative signs is even or odd.

• If the number of negative integers being multiplied is even (like 2, 4, 6, ...), the product will be a positive integer. (Pairs of negative signs cancel out to positive: $(-)\times(-) = (+)$).
• If the number of negative integers being multiplied is odd (like 1, 3, 5, ...), the product will be a negative integer. (After pairing up negative signs to get positives, one negative sign will remain).

Examples:

1. $(-2) \times (-3) \times (-4)$

There are three negative integers in the product (3 is an odd number). So the final product will be negative.

Let's calculate step-by-step: First multiply $(-2) \times (-3) = +(2 \times 3) = 6$.

Now multiply the result by $(-4)$: $6 \times (-4) = -(6 \times 4) = -24$.

2. $(-2) \times (-3) \times (-4) \times (-1)$

There are four negative integers in the product (4 is an even number). So the final product will be positive.

Calculate step-by-step (or group): $(-2) \times (-3) = 6$.

$(-4) \times (-1) = +(4 \times 1) = 4$.

Now multiply the results: $6 \times 4 = 24$.

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Division of Integers

We have explored addition, subtraction, and multiplication of integers. Now, let's understand Division of Integers. Division is the inverse operation of multiplication. This means that division 'undoes' multiplication. If we know that the product of two integers $a$ and $b$ is $c$ (i.e., $a \times b = c$), then dividing $c$ by $a$ gives $b$ (i.e., $c \div a = b$, provided $a \neq 0$), and dividing $c$ by $b$ gives $a$ (i.e., $c \div b = a$, provided $b \neq 0$). Division helps us split a quantity into equal parts or determine how many times one quantity is contained within another.

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Rules for Division of Integers

Just like multiplication, the sign of the quotient (the result of division) depends on the signs of the integers being divided (the dividend and the divisor). The sign rules for division are directly related to the sign rules for multiplication because of the inverse relationship between the two operations.

Let $a$ and $b$ be non-zero integers.

1. Division of two positive integers

The quotient of two positive integers is a positive integer. This is the same as division of whole numbers.

Example: $(+10) \div (+2)$. Since $(+2) \times (+5) = +10$, then $(+10) \div (+2) = +5$, or simply $5$.

Example: $24 \div 6 = 4$.

2. Division of two negative integers

The quotient of two negative integers is a positive integer. To find the quotient, divide their absolute values.

Example: $(-10) \div (-2)$. We need a number that, when multiplied by $-2$, gives $-10$. We know $(-2) \times (+5) = -10$. So, the quotient is $+5$, or $5$.

Example: $(-36) \div (-4) = +(36 \div 4) = 9$ (Since $(-4) \times 9 = -36$).

3. Division of one positive and one negative integer (or vice-versa)

The quotient of a positive integer and a negative integer (in either order) is always a negative integer. To find the quotient, divide their absolute values and place a negative sign before the result.

Example 1: $(+10) \div (-2)$. We need a number that, when multiplied by $-2$, gives $+10$. We know $(-2) \times (-5) = 10$. So, the quotient is $-5$.

Example 2: $(-10) \div (+2)$. We need a number that, when multiplied by $+2$, gives $-10$. We know $(+2) \times (-5) = -10$. So, the quotient is $-5$.

Example 3: $45 \div (-9) = -(45 \div 9) = -5$ (Since $(-9) \times (-5) = 45$).

Example 4: $(-28) \div 7 = -(28 \div 7) = -4$ (Since $7 \times (-4) = -28$).

Summary of Sign Rules for Division:

The sign rules for division are the same as those for multiplication:

Dividend Sign	Divisor Sign	Quotient Sign
Positive (+)	Positive (+)	Positive (+)
Negative (-)	Negative (-)	Positive (+)
Positive (+)	Negative (-)	Negative (-)
Negative (-)	Positive (+)	Negative (-)

In short: If the dividend and divisor have the **same sign**, the quotient is **positive**. If they have **different signs**, the quotient is **negative**.

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Properties of Division of Integers

While multiplication of integers has many properties like closure, commutativity, and associativity, division of integers does not hold these properties in general.

1. Closure Property

The set of integers is not closed under division. This means that when you divide one integer by another integer, the result is not always an integer.

Example: $5 \div 2 = \frac{5}{2} = 2.5$. This is a decimal number, which is not an integer.

Example: $(-7) \div 3 = -\frac{7}{3}$. This is a fraction, which is not an integer.

The result of dividing two integers is a rational number. It will be an integer only if the dividend is perfectly divisible by the divisor (i.e., the remainder is zero).

2. Commutative Property

Division of integers is not commutative. Changing the order of the integers in a division problem generally changes the result.

In mathematical terms: For integers $a$ and $b$ ($b \neq 0, a \neq b$), $a \div b \neq b \div a$.

Example:

Let $a = -8$ and $b = 4$.

$a \div b = (-8) \div 4 = -2$.

$b \div a = 4 \div (-8) = \frac{4}{-8} = -\frac{1}{2}$.

Since $-2 \neq -\frac{1}{2}$, division is not commutative for integers.

3. Associative Property

Division of integers is not associative. The way you group integers when performing multiple divisions affects the result.

In mathematical terms: For integers $a, b, c$ ($b \neq 0, c \neq 0$), $(a \div b) \div c \neq a \div (b \div c)$ generally.

Example:

Let $a = -16$, $b = 4$, $c = -2$.

Calculate the Left Hand Side (LHS): $(a \div b) \div c$

$(a \div b) \div c = ((-16) \div 4) \div (-2)$

First, calculate inside the parenthesis: $(-16) \div 4$. (Negative $\div$ Positive $=$ Negative). $(-16) \div 4 = -4$.

Now, divide the result by $c$: $(-4) \div (-2)$. (Negative $\div$ Negative $=$ Positive). $(-4) \div (-2) = 2$.

So, LHS $= 2$.

Calculate the Right Hand Side (RHS): $a \div (b \div c)$

$a \div (b \div c) = (-16) \div (4 \div (-2))$

First, calculate inside the parenthesis: $4 \div (-2)$. (Positive $\div$ Negative $=$ Negative). $4 \div (-2) = -2$.

Now, divide $a$ by the result: $(-16) \div (-2)$. (Negative $\div$ Negative $=$ Positive). $(-16) \div (-2) = 8$.

So, RHS $= 8$.

Since LHS ($2$) is not equal to RHS ($8$), division is not associative for integers.

4. Property of Zero in Division

Zero has specific properties in division:

(a) Division of zero by any non-zero integer is zero.

If '$a$' is any non-zero integer ($a \neq 0$), then $0 \div a = 0$. This is because $a \times 0 = 0$.

Example: $0 \div 5 = 0$. $0 \div (-3) = 0$. $0 \div 100 = 0$.

(b) Division of any integer by zero is undefined.

Dividing any integer by $0$ is not defined in mathematics. This is because there is no number that, when multiplied by 0, gives a non-zero result. Also, if we were to define $a \div 0 = b$, it would imply $b \times 0 = a$. If $a \neq 0$, this is impossible. If $a=0$, then $b \times 0 = 0$, which is true for any value of $b$, so the result is not a unique number.

If '$a$' is any integer, $a \div 0$ is undefined.

Example: $7 \div 0$ is undefined. $(-4) \div 0$ is undefined.

5. Property of One in Division

One also has specific properties in division:

(a) Any integer divided by 1 gives the integer itself. This is the multiplicative identity property in reverse.

If '$a$' is any integer, then $a \div 1 = a$.

Example: $8 \div 1 = 8$. $(-5) \div 1 = -5$. $0 \div 1 = 0$.

(b) Any non-zero integer divided by itself gives 1.

If '$a$' is any non-zero integer ($a \neq 0$), then $a \div a = 1$. This is consistent with the rule that dividing two numbers with the same sign gives a positive result.

Example: $8 \div 8 = 1$. $(-5) \div (-5) = 1$.

(c) Any non-zero integer divided by -1 gives its additive inverse.

If '$a$' is any non-zero integer ($a \neq 0$), then $a \div (-1) = -a$. This is consistent with the rule that dividing two numbers with different signs gives a negative result.

Example: $8 \div (-1) = -8$ (additive inverse of 8). $(-5) \div (-1) = 5$ (additive inverse of -5).

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Order of Performing Operations

In mathematics, expressions can sometimes involve multiple operations, such as addition, subtraction, multiplication, division, and even brackets. When you have an expression with more than one operation, the order in which you perform these operations is crucial to get the correct answer. If you don't follow a specific order, you might get a different result each time, which would be confusing! To ensure consistency and accuracy, we follow a standard rule for the order of operations. This rule is commonly remembered by the acronym BODMAS (or sometimes PEMDAS in other countries).

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The BODMAS Rule

The acronym BODMAS tells us the sequence in which operations should be performed in a mathematical expression:

• B - Brackets: First, solve the operations inside any brackets. If there are different types of brackets (like parentheses, curly brackets, and square brackets), you should solve the innermost brackets first and work outwards.
• O - Orders: Next, evaluate any powers (like $2^2$, $3^3$) or roots (like $\sqrt{9}$). In some contexts, 'O' might also stand for 'Of', which usually indicates multiplication (especially used with fractions, e.g., $\frac{1}{2}$ of 10 means $\frac{1}{2} \times 10$).
• D - Division: After dealing with brackets and orders, perform any division operations.
• M - Multiplication: Next, perform any multiplication operations.
• A - Addition: After division and multiplication, perform any addition operations.
• S - Subtraction: Finally, perform any subtraction operations.

The operations at the same level of precedence (like Division and Multiplication, or Addition and Subtraction) should be performed from left to right as they appear in the expression.

• Division and Multiplication are at the same level. Work from left to right. Example: $10 \div 2 \times 5$. Division comes first from the left ($10 \div 2 = 5$), then multiplication ($5 \times 5 = 25$).
• Addition and Subtraction are at the same level. Work from left to right. Example: $10 - 5 + 3$. Subtraction comes first from the left ($10 - 5 = 5$), then addition ($5 + 3 = 8$).

Types of Brackets and Order of Simplification:

If an expression contains multiple types of brackets, there's a specific order in which they should be simplified, starting from the innermost bracket and moving outwards:

1. Vinculum or Bar ($ \overline{\phantom{aa}} $): This is a line drawn over a part of the expression. It's treated as the innermost grouping symbol and should be simplified first if present. Example: $5 \times \overline{4-2} = 5 \times 2 = 10$.
2. Parentheses or Round Brackets ($ ( ) $): These are the next brackets to be simplified after any vinculum.
3. Curly Brackets or Braces ($ \{ \} $): These are simplified after round brackets.
4. Square Brackets or Box Brackets ($ [ ] $): These are the outermost brackets and are simplified last.

The order of removing brackets is: Vinculum $\to$ Parentheses $\to$ Curly Brackets $\to$ Square Brackets.

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Examples using BODMAS Rule

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